We investigate the power of randomness in two-party communication complexity. In particular, we study the model where the parties can make a constant number of queries to a function that has an efficient one-sided-error randomized protocol. The complexity classes defined by this model comprise the Randomized Boolean Hierarchy, which is analogous to the Boolean Hierarchy but defined with one-sidederror randomness instead of nondeterminism. Our techniques connect the Nondeterministic and Randomized Boolean Hierarchies, and we provide a complete picture of the relationships among complexity classes within and across these two hierarchies. In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it to resolve an open problem stated in the paper by Halstenberg and Reischuk (CCC 1988) which initiated the study of this hierarchy.

我们研究了两方通信复杂性中随机性的力量。特别是，我们研究了一种模型，其中各方可以对具有高效单边错误随机协议的函数进行恒定数量的查询。该模型定义的复杂性类别包括随机布尔层次结构，它类似于布尔层次结构，但定义为单边误差随机性而不是不确定性。我们的技术连接了非确定性和随机布尔层次结构，并且我们提供了这两个层次结构内部和之间的复杂性类之间关系的完整图片。特别是，我们证明了随机布尔层次结构不会崩溃，并且我们证明了非确定性布尔层次结构所有级别的查询到通信提升定理，并用它来解决 Halstenberg 和 Reischuk 论文中提出的开放问题（ CCC 1988）发起了对这种层次结构的研究。